/export/starexec/sandbox/solver/bin/starexec_run_ttt2-1.17+nonreach /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


--------------------------------------------------------------------------------


YES

Problem:
 app(f(),app(g(),x)) -> app(g(),app(f(),app(f(),x)))
 app(f(),app(h(),x)) -> app(h(),app(g(),x))
 app(app(map(),fun),nil()) -> nil()
 app(app(map(),fun),app(app(cons(),x),xs)) -> app(app(cons(),app(fun,x)),app(app(map(),fun),xs))
 app(app(filter(),fun),nil()) -> nil()
 app(app(filter(),fun),app(app(cons(),x),xs)) -> app(app(app(app(filter2(),app(fun,x)),fun),x),xs)
 app(app(app(app(filter2(),true()),fun),x),xs) -> app(app(cons(),x),app(app(filter(),fun),xs))
 app(app(app(app(filter2(),false()),fun),x),xs) -> app(app(filter(),fun),xs)

Proof:
 Extended Uncurrying Processor:
  application symbol: app
  symbol table:
   false ==> false0/0
   true ==> true0/0
   filter2 ==> filter20/0 filter21/1 filter22/2 filter23/3 filter24/4
   filter ==> filter0/0 filter1/1 filter2/2
   cons ==> cons0/0 cons1/1 cons2/2
   nil ==> nil0/0
   map ==> map0/0 map1/1 map2/2
   h ==> h0/0 h1/1
   g ==> g0/0 g1/1
   f ==> f0/0 f1/1
   
  uncurry-rules:
   app(f0(),x3) -> f1(x3)
   app(g0(),x5) -> g1(x5)
   app(h0(),x7) -> h1(x7)
   app(map1(x9),x10) -> map2(x9,x10)
   app(map0(),x9) -> map1(x9)
   app(cons1(x13),x14) -> cons2(x13,x14)
   app(cons0(),x13) -> cons1(x13)
   app(filter1(x16),x17) -> filter2(x16,x17)
   app(filter0(),x16) -> filter1(x16)
   app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
   app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
   app(filter21(x19),x20) -> filter22(x19,x20)
   app(filter20(),x19) -> filter21(x19)
  eta-rules:
   
  problem:
   f1(g1(x)) -> g1(f1(f1(x)))
   f1(h1(x)) -> h1(g1(x))
   map2(fun,nil0()) -> nil0()
   map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
   filter2(fun,nil0()) -> nil0()
   filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
   filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
   filter24(false0(),fun,x,xs) -> filter2(fun,xs)
   app(f0(),x3) -> f1(x3)
   app(g0(),x5) -> g1(x5)
   app(h0(),x7) -> h1(x7)
   app(map1(x9),x10) -> map2(x9,x10)
   app(map0(),x9) -> map1(x9)
   app(cons1(x13),x14) -> cons2(x13,x14)
   app(cons0(),x13) -> cons1(x13)
   app(filter1(x16),x17) -> filter2(x16,x17)
   app(filter0(),x16) -> filter1(x16)
   app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
   app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
   app(filter21(x19),x20) -> filter22(x19,x20)
   app(filter20(),x19) -> filter21(x19)
  DP Processor:
   DPs:
    f{1,#}(g1(x)) -> f{1,#}(x)
    f{1,#}(g1(x)) -> f{1,#}(f1(x))
    map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
    map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
    filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
    filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
    filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
    filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
    app#(f0(),x3) -> f{1,#}(x3)
    app#(map1(x9),x10) -> map{2,#}(x9,x10)
    app#(filter1(x16),x17) -> filter{2,#}(x16,x17)
    app#(filter23(x19,x20,x21),x22) -> filter2{4,#}(x19,x20,x21,x22)
   TRS:
    f1(g1(x)) -> g1(f1(f1(x)))
    f1(h1(x)) -> h1(g1(x))
    map2(fun,nil0()) -> nil0()
    map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
    filter2(fun,nil0()) -> nil0()
    filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
    filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
    filter24(false0(),fun,x,xs) -> filter2(fun,xs)
    app(f0(),x3) -> f1(x3)
    app(g0(),x5) -> g1(x5)
    app(h0(),x7) -> h1(x7)
    app(map1(x9),x10) -> map2(x9,x10)
    app(map0(),x9) -> map1(x9)
    app(cons1(x13),x14) -> cons2(x13,x14)
    app(cons0(),x13) -> cons1(x13)
    app(filter1(x16),x17) -> filter2(x16,x17)
    app(filter0(),x16) -> filter1(x16)
    app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
    app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
    app(filter21(x19),x20) -> filter22(x19,x20)
    app(filter20(),x19) -> filter21(x19)
   TDG Processor:
    DPs:
     f{1,#}(g1(x)) -> f{1,#}(x)
     f{1,#}(g1(x)) -> f{1,#}(f1(x))
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(f0(),x3) -> f{1,#}(x3)
     app#(map1(x9),x10) -> map{2,#}(x9,x10)
     app#(filter1(x16),x17) -> filter{2,#}(x16,x17)
     app#(filter23(x19,x20,x21),x22) -> filter2{4,#}(x19,x20,x21,x22)
    TRS:
     f1(g1(x)) -> g1(f1(f1(x)))
     f1(h1(x)) -> h1(g1(x))
     map2(fun,nil0()) -> nil0()
     map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
     filter2(fun,nil0()) -> nil0()
     filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
     filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
     filter24(false0(),fun,x,xs) -> filter2(fun,xs)
     app(f0(),x3) -> f1(x3)
     app(g0(),x5) -> g1(x5)
     app(h0(),x7) -> h1(x7)
     app(map1(x9),x10) -> map2(x9,x10)
     app(map0(),x9) -> map1(x9)
     app(cons1(x13),x14) -> cons2(x13,x14)
     app(cons0(),x13) -> cons1(x13)
     app(filter1(x16),x17) -> filter2(x16,x17)
     app(filter0(),x16) -> filter1(x16)
     app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
     app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
     app(filter21(x19),x20) -> filter22(x19,x20)
     app(filter20(),x19) -> filter21(x19)
    graph:
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs) ->
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x19,x20,x21),x22) -> filter2{4,#}(x19,x20,x21,x22)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x16),x17) -> filter{2,#}(x16,x17)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x9),x10) -> map{2,#}(x9,x10)
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f0(),x3) -> f{1,#}(x3)
     app#(filter23(x19,x20,x21),x22) -> filter2{4,#}(x19,x20,x21,x22) ->
     filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter23(x19,x20,x21),x22) -> filter2{4,#}(x19,x20,x21,x22) ->
     filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
     app#(filter1(x16),x17) -> filter{2,#}(x16,x17) ->
     filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
     app#(filter1(x16),x17) -> filter{2,#}(x16,x17) ->
     filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x9),x10) -> map{2,#}(x9,x10) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     app#(map1(x9),x10) -> map{2,#}(x9,x10) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     app#(f0(),x3) -> f{1,#}(x3) -> f{1,#}(g1(x)) -> f{1,#}(f1(x))
     app#(f0(),x3) -> f{1,#}(x3) ->
     f{1,#}(g1(x)) -> f{1,#}(x)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter23(x19,x20,x21),x22) -> filter2{4,#}(x19,x20,x21,x22)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(filter1(x16),x17) -> filter{2,#}(x16,x17)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(map1(x9),x10) -> map{2,#}(x9,x10)
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x) ->
     app#(f0(),x3) -> f{1,#}(x3)
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs) ->
     map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs) ->
     map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
     f{1,#}(g1(x)) -> f{1,#}(f1(x)) -> f{1,#}(g1(x)) -> f{1,#}(f1(x))
     f{1,#}(g1(x)) -> f{1,#}(f1(x)) -> f{1,#}(g1(x)) -> f{1,#}(x)
     f{1,#}(g1(x)) -> f{1,#}(x) -> f{1,#}(g1(x)) -> f{1,#}(f1(x))
     f{1,#}(g1(x)) -> f{1,#}(x) -> f{1,#}(g1(x)) -> f{1,#}(x)
    SCC Processor:
     #sccs: 2
     #rules: 11
     #arcs: 28/144
     DPs:
      filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
      filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(map1(x9),x10) -> map{2,#}(x9,x10)
      map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
      map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
      app#(filter1(x16),x17) -> filter{2,#}(x16,x17)
      filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
      filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
      app#(filter23(x19,x20,x21),x22) -> filter2{4,#}(x19,x20,x21,x22)
     TRS:
      f1(g1(x)) -> g1(f1(f1(x)))
      f1(h1(x)) -> h1(g1(x))
      map2(fun,nil0()) -> nil0()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil0()) -> nil0()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false0(),fun,x,xs) -> filter2(fun,xs)
      app(f0(),x3) -> f1(x3)
      app(g0(),x5) -> g1(x5)
      app(h0(),x7) -> h1(x7)
      app(map1(x9),x10) -> map2(x9,x10)
      app(map0(),x9) -> map1(x9)
      app(cons1(x13),x14) -> cons2(x13,x14)
      app(cons0(),x13) -> cons1(x13)
      app(filter1(x16),x17) -> filter2(x16,x17)
      app(filter0(),x16) -> filter1(x16)
      app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
      app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
      app(filter21(x19),x20) -> filter22(x19,x20)
      app(filter20(),x19) -> filter21(x19)
     Subterm Criterion Processor:
      simple projection:
       pi(map{2,#}) = 0
       pi(app#) = 0
       pi(filter{2,#}) = 0
       pi(filter2{4,#}) = 1
      problem:
       DPs:
        filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
        filter{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
        map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
        map{2,#}(fun,cons2(x,xs)) -> app#(fun,x)
        filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
        filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
       TRS:
        f1(g1(x)) -> g1(f1(f1(x)))
        f1(h1(x)) -> h1(g1(x))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(f0(),x3) -> f1(x3)
        app(g0(),x5) -> g1(x5)
        app(h0(),x7) -> h1(x7)
        app(map1(x9),x10) -> map2(x9,x10)
        app(map0(),x9) -> map1(x9)
        app(cons1(x13),x14) -> cons2(x13,x14)
        app(cons0(),x13) -> cons1(x13)
        app(filter1(x16),x17) -> filter2(x16,x17)
        app(filter0(),x16) -> filter1(x16)
        app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
        app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
        app(filter21(x19),x20) -> filter22(x19,x20)
        app(filter20(),x19) -> filter21(x19)
      SCC Processor:
       #sccs: 2
       #rules: 4
       #arcs: 20/36
       DPs:
        map{2,#}(fun,cons2(x,xs)) -> map{2,#}(fun,xs)
       TRS:
        f1(g1(x)) -> g1(f1(f1(x)))
        f1(h1(x)) -> h1(g1(x))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(f0(),x3) -> f1(x3)
        app(g0(),x5) -> g1(x5)
        app(h0(),x7) -> h1(x7)
        app(map1(x9),x10) -> map2(x9,x10)
        app(map0(),x9) -> map1(x9)
        app(cons1(x13),x14) -> cons2(x13,x14)
        app(cons0(),x13) -> cons1(x13)
        app(filter1(x16),x17) -> filter2(x16,x17)
        app(filter0(),x16) -> filter1(x16)
        app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
        app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
        app(filter21(x19),x20) -> filter22(x19,x20)
        app(filter20(),x19) -> filter21(x19)
       Subterm Criterion Processor:
        simple projection:
         pi(map{2,#}) = 1
        problem:
         DPs:
          
         TRS:
          f1(g1(x)) -> g1(f1(f1(x)))
          f1(h1(x)) -> h1(g1(x))
          map2(fun,nil0()) -> nil0()
          map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
          filter2(fun,nil0()) -> nil0()
          filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
          filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
          filter24(false0(),fun,x,xs) -> filter2(fun,xs)
          app(f0(),x3) -> f1(x3)
          app(g0(),x5) -> g1(x5)
          app(h0(),x7) -> h1(x7)
          app(map1(x9),x10) -> map2(x9,x10)
          app(map0(),x9) -> map1(x9)
          app(cons1(x13),x14) -> cons2(x13,x14)
          app(cons0(),x13) -> cons1(x13)
          app(filter1(x16),x17) -> filter2(x16,x17)
          app(filter0(),x16) -> filter1(x16)
          app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
          app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
          app(filter21(x19),x20) -> filter22(x19,x20)
          app(filter20(),x19) -> filter21(x19)
        Qed
       
       DPs:
        filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
        filter{2,#}(fun,cons2(x,xs)) -> filter2{4,#}(app(fun,x),fun,x,xs)
        filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
       TRS:
        f1(g1(x)) -> g1(f1(f1(x)))
        f1(h1(x)) -> h1(g1(x))
        map2(fun,nil0()) -> nil0()
        map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
        filter2(fun,nil0()) -> nil0()
        filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
        filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
        filter24(false0(),fun,x,xs) -> filter2(fun,xs)
        app(f0(),x3) -> f1(x3)
        app(g0(),x5) -> g1(x5)
        app(h0(),x7) -> h1(x7)
        app(map1(x9),x10) -> map2(x9,x10)
        app(map0(),x9) -> map1(x9)
        app(cons1(x13),x14) -> cons2(x13,x14)
        app(cons0(),x13) -> cons1(x13)
        app(filter1(x16),x17) -> filter2(x16,x17)
        app(filter0(),x16) -> filter1(x16)
        app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
        app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
        app(filter21(x19),x20) -> filter22(x19,x20)
        app(filter20(),x19) -> filter21(x19)
       Subterm Criterion Processor:
        simple projection:
         pi(filter{2,#}) = 1
         pi(filter2{4,#}) = 3
        problem:
         DPs:
          filter2{4,#}(false0(),fun,x,xs) -> filter{2,#}(fun,xs)
          filter2{4,#}(true0(),fun,x,xs) -> filter{2,#}(fun,xs)
         TRS:
          f1(g1(x)) -> g1(f1(f1(x)))
          f1(h1(x)) -> h1(g1(x))
          map2(fun,nil0()) -> nil0()
          map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
          filter2(fun,nil0()) -> nil0()
          filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
          filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
          filter24(false0(),fun,x,xs) -> filter2(fun,xs)
          app(f0(),x3) -> f1(x3)
          app(g0(),x5) -> g1(x5)
          app(h0(),x7) -> h1(x7)
          app(map1(x9),x10) -> map2(x9,x10)
          app(map0(),x9) -> map1(x9)
          app(cons1(x13),x14) -> cons2(x13,x14)
          app(cons0(),x13) -> cons1(x13)
          app(filter1(x16),x17) -> filter2(x16,x17)
          app(filter0(),x16) -> filter1(x16)
          app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
          app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
          app(filter21(x19),x20) -> filter22(x19,x20)
          app(filter20(),x19) -> filter21(x19)
        SCC Processor:
         #sccs: 0
         #rules: 0
         #arcs: 4/4
         
     
     DPs:
      f{1,#}(g1(x)) -> f{1,#}(x)
      f{1,#}(g1(x)) -> f{1,#}(f1(x))
     TRS:
      f1(g1(x)) -> g1(f1(f1(x)))
      f1(h1(x)) -> h1(g1(x))
      map2(fun,nil0()) -> nil0()
      map2(fun,cons2(x,xs)) -> cons2(app(fun,x),map2(fun,xs))
      filter2(fun,nil0()) -> nil0()
      filter2(fun,cons2(x,xs)) -> filter24(app(fun,x),fun,x,xs)
      filter24(true0(),fun,x,xs) -> cons2(x,filter2(fun,xs))
      filter24(false0(),fun,x,xs) -> filter2(fun,xs)
      app(f0(),x3) -> f1(x3)
      app(g0(),x5) -> g1(x5)
      app(h0(),x7) -> h1(x7)
      app(map1(x9),x10) -> map2(x9,x10)
      app(map0(),x9) -> map1(x9)
      app(cons1(x13),x14) -> cons2(x13,x14)
      app(cons0(),x13) -> cons1(x13)
      app(filter1(x16),x17) -> filter2(x16,x17)
      app(filter0(),x16) -> filter1(x16)
      app(filter23(x19,x20,x21),x22) -> filter24(x19,x20,x21,x22)
      app(filter22(x19,x20),x21) -> filter23(x19,x20,x21)
      app(filter21(x19),x20) -> filter22(x19,x20)
      app(filter20(),x19) -> filter21(x19)
     Usable Rule Processor:
      DPs:
       f{1,#}(g1(x)) -> f{1,#}(x)
       f{1,#}(g1(x)) -> f{1,#}(f1(x))
      TRS:
       f1(g1(x)) -> g1(f1(f1(x)))
       f1(h1(x)) -> h1(g1(x))
      Semantic Labeling Processor:
       dimension: 1
       usable rules:
        
       interpretation:
        [h1](x0) = 0,
        
        [g1](x0) = x0 + 1,
        
        [f1](x0) = x0
        labeled:
         f{1,#}
        usable (for model):
         f{1,#}
         g1
         f1
         h1
        argument filtering:
         pi(f1) = 0
         pi(g1) = 0
         pi(h1) = []
         pi(f{1,#}) = []
       precedence:
        f{1,#} ~ h1 ~ g1 ~ f1
       problem:
        DPs:
         
        TRS:
         f1(g1(x)) -> g1(f1(f1(x)))
         f1(h1(x)) -> h1(g1(x))
       Qed